Ex - Balance Scale

AtCoder

IDabc306_h

Time3000ms

Memory256MB

Difficulty—

There are $N$ weights numbered $1,2, \dots,N$. Using a balance, we will compare weights $M$ times. * Before the comparisons, prepare an empty string $S$. * For the $i$\-th comparison, put just weight $A_i$ to the left bowl, and just weight $B_i$ to the right. * Then, one of the following three results is obtained. * If weight $A_i$ is heavier than weight $B_i$, * append `>` to the tail of $S$. * If weight $A_i$ and weight $B_i$ have the same mass, * append `=` to the tail of $S$. * If weight $A_i$ is lighter than weight $B_i$, * append `<` to the tail of $S$. * The result is always accurate. After the experiment, you will obtain a string $S$ of length $M$. Among the $3^M$ strings of length $M$ consisting of `>`, `=`, and `<`, how many can be obtained as $S$ by the experiment? Since the answer can be enormous, print the answer modulo $998244353$. ## Constraints * All input values are integers. * $2 \le N \le 17$ * $1 \le M \le \frac{N \times (N-1)}{2}$ * $1 \le A_i < B_i \le N$ * $i \neq j \Rightarrow (A_i,B_i) \neq (A_j,B_j)$ ## Input The input is given from Standard Input in the following format: $N$ $M$ $A_1$ $B_1$ $A_2$ $B_2$ $\vdots$ $A_M$ $B_M$ [samples]

Samples

Input #1

Output #1

13

Let $w$ be the sequence of the mass of the weights, in ascending order of weight numbers.

*   If $w=(5,5,5)$, you obtain $S=$ `===`.
*   If $w=(2,2,3)$, you obtain $S=$ `=<<`.
*   If $w=(6,8,6)$, you obtain $S=$ `<=>`.
*   If $w=(9,4,4)$, you obtain $S=$ `>>=`.
*   If $w=(7,7,3)$, you obtain $S=$ `=>>`.
*   If $w=(8,1,8)$, you obtain $S=$ `>=<`.
*   If $w=(5,8,8)$, you obtain $S=$ `<<=`.
*   If $w=(1,2,3)$, you obtain $S=$ `<<<`.
*   If $w=(4,9,5)$, you obtain $S=$ `<<>`.
*   If $w=(5,1,8)$, you obtain $S=$ `><<`.
*   If $w=(6,9,2)$, you obtain $S=$ `<>>`.
*   If $w=(7,1,3)$, you obtain $S=$ `>><`.
*   If $w=(9,7,5)$, you obtain $S=$ `>>>`.

While there is an infinite number of possible sequences of the mass of the weights, $S$ is always one of the $13$ above.

Input #2

Output #2

Input #3

Output #3

API Response (JSON)

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    "name": "Ex - Balance Scale",
    "description": {
      "content": "There are $N$ weights numbered $1,2, \\dots,N$.   Using a balance, we will compare weights $M$ times. *   Before the comparisons, prepare an empty string $S$. *   For the $i$\\-th comparison, put just ",
      "description_type": "Markdown"
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    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
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      "statement_type": "Markdown",
      "content": "There are $N$ weights numbered $1,2, \\dots,N$.  \nUsing a balance, we will compare weights $M$ times.\n\n*   Before the comparisons, prepare an empty string $S$.\n*   For the $i$\\-th comparison, put just ...",
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      "language": "English"
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